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Degenerate Fourier integral operators in the plane. (English) Zbl 0806.35191

Soient X et Y deux ouverts bornés de 2 , une hypersurface dans X×Y et N * son fibré normal. On pose Λ=N * 0 et on fait l’hypothèse que Λ(T * X0)×(T * Y0). On considère x ={yY;(x,y)} et ψC 0 (X×Y). Si dσ n est une densité régulière sur x , dépendant régulièrement de x on définit l’opérateur

Rf(x)= x ψ(x,y)f(y)dσ x (y)

qui est un FIO de classe I -1/2 (X,Y,Λ). Si Λ est localement le graph d’une transformation canonique (donc si les projections π R :ΛT * Y et π L :ΛT * X sont localement des difféomorphismes), l’opérateur R a des propriétés de continuité L p L q et L p L α p bien connues. L’auteur prouve de telles propriétés dans le cas où π L , π R sont singulières.

35S30Fourier integral operators